Publicly provided disaster insurance for health and the control of moral hazard

Journal of Public Economics 39 (1989) 141-156. North-Holland

PUBLICLY PROVIDED DISASTER INSURANCE FOR HEALTH AND THE CONTROL OF MORAL HAZARD

Timothy BESLEY*

Woodrow Wilson School, Princeton Unioersity, Princeton, NJ 08544, USA

Received August 1988, revised version received March 1989

This paper looks at the moral hazard due to health insurance offering a subsidy to the consumption of health care at the margin. We show how the institution of publicly provided disaster insurance yields a welfare improvement. This result is obtained since public insurance encourages insured individuals to reduce the amount of private insurance that they buy and hence diminishes the moral hazard problem.

1. Introduction

In designing schemes for dealing with ill-health, one has in mind many concerns, among them the trade-off between risk sharing and incentives.’ Given that adverse selection is largely dealt with by using group schemes for insurance coverage, it is most natural, in the context of health insurance, to focus upon the moral hazard problem. In the literature on medical insurance, the term moral hazard has come to stand for a multitude of sins, although the main agreed symptom is health care expenditure which is excessive in some appropriately defined sense. The three main sources of moral hazard are: (i) the decreased incentive of the insured to take care once she has insurance; (ii) the fact that physicians may prescribe more elaborate treat- ments than are strictly necessary; and (iii) the subsidy given to health care at the margin.

This paper focuses upon the third type of moral hazard problem and the possible role of publicly provided insurance for very severe illnesses in combating it. To make the focus as sharp as possible, we use a model of reimbursement insurance. The conclusions that we draw from this case may conflict with those reached in the analysis of different moral hazard

*This paper is based on parts of my Oxford D.Phil. thesis (1987). I am grateful to Terence Gorman, David Newbery, Christopher Gilbert and to two anonymous referees for helpful comments and advice. Remaining errors are my responsibility.

‘There are a number of papers which have already examined aspects of it [see, inter alia, Zeckhauser (1970), Feldstein (1973), Pauly (1968, 1974), Holmstrom (1979) and Besley (1988)].

0047-2727/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

142 7: Besley, Publicly provided disaster insurance

problems.’ In spite of this, the piecemeal approach adopted here still serves a purpose in gaining a better understanding of the problem at hand. We consider a world in which the market for insurance is competitive. Pauly (1974) emphasized that in such markets it may be impossible to operate premium functions which are increasing in the level of coverage chosen, since individuals have the option of purchasing insurance coverage from a number of different insurers3 Here, we follow Pauly in maintaining the assumption that the price per unit of coverage (the premium) does not increase with the amount of coverage taken.

Using publicly provided insurance to combat moral hazard is not a new idea [see, for example, Feldstein (1977)]. This paper shows just why the intuitive argument in favor of it is justified. It is also an issue of practical relevance. It figures in discussions of how best to organize the predominantly market-oriented U.S. system of health care provision, and in the United Kingdom, which is witnessing an expansion of private health insurance, there is much debate about the proper function of the state system. In any of these cases, moral hazard is only part of the story, although possibly an important

part. The structure of this paper is as follows. Section 2 lays out a formal model

of the demand for health care and health insurance. In section 3 we demonstrate, using this model, that the introduction of public health insurance to deal with very severe (in the sense of high expenditure) outcomes may raise welfare. Section 4 concludes.

2. A model

Consider a single consumer, facing prices determined in perfectly competitive markets, who consumes two types of goods: ‘health goods’ and ‘non- health goods’. These are denoted respectively, by XE R, and ZE R,. The corresponding price vectors are p and q. We also introduce a good which we refer to as ‘health’ denoted by YE Rf. Utility depends upon non-health goods

and health, namely

u=U(z,.!J),

which satisfies:

(2.1)

‘Mirrlees (1979) showed that moral hazard, due to a reduction in the amount of care taken, may require an optimal response of offering no insurance against severe outcomes in the case of a monopoly insurer, This is so since such a scheme may give the insured the best incentive to take care.

‘He remarks that ‘the insurer knows how much insurance he has sold the individual, but he does not know how much insurance the individual has bought from others’ [Pauly (1974, p. 50) (emphasis in original)].

7: Besley, Publicly provided disaster insurance 143

Assumption 1. U: R,+,-+R + is strictly concave and strictly increasing.

Health is produced from health goods given a pre-treatment health state (which can be thought of as referring to a state of illness) denoted by 0 E [w+ with the technology given by

4’ =g(x, 0). (2.2)

This technology is assumed to satisfy:

Assumption 2. g: iw, + 1 + R + is increasing in each element of x, decreasing in H and quasi-homothetic, i.e. has a cost function of the form:

4p, 6 Y) = min {p. .x 1 g(x, 0) 2 Y) = 4p, 0)~ + Wp, 0). X

(2.3)

The assumption of quasi-homotheticity is particularly convenient.4 It permits us to talk of the demand for health since decentralized budgeting in the sense of Strotz (1959) is permissible. Hence, our model is very much in the spirit of Grossman (1972) although it differs substantially in its analytical structure. The notion of decentralization is also agreeable if one considers that choices over health goods might be undertaken by physicians. In the analysis of this paper we do not wish to consider the role of physicians directly. However, they can be thought of as controlling the technology g(.;) and hence influencing the demand for health care in this way.

The two parts of the cost function have straightforward interpretations. The function u(p, 0) is the marginal cost of health, and b(p,O) is the fixed cost. The former is also the price of health in a meaningful sense. Assumption 2 ensures that these functions are increasing in 8. The indirect utility function associated with (2.1) is given by:

where I is lump-sum income. We shall also assume:

Assumption 3. V: R, + 2 -+ R ’ is a twice continuously differentiable function of each of its arguments.

Above, we suggested that we could treat a(~,@ as a price argument in the indirect utility function. Using Roy’s Identity in this vein yields:

4We owe this terminology to Gorman; see, for example, Gorman (1976).


y=y(a(p, e), q, W(p, I, f3)) = - aV(*)‘au aV(.)/aw’

(2.5)

where we have defined, W(p,l,d)-I- b(p,O). This yields the demand for health and reveals the precise sense in which a(p,O) is the price of health. Differentiating (2.4) with respect to the price of typical health good i yields, after using (2.5):

xi= - aqyaw wp, 4 aVC)‘api = BaDpbs) y(~(p, e), q, wtp, I, e)) + ~ , 34 ’ (2.6)

which is the demand for health good i. Quasi-homotheticity ensures that (2.6) depends upon the price of non-health goods and income only through changes in the demand for health.

In the sequel, we use y to denote the quantity of health, i.e. a real number, and y to denote health as a function of appropriate parameters, i.e. a real valued function. The importance of this distinction can be seen by noting that e(p, 0, y) is an increasing function of 8 (by Assumption 2), while e(p, 8, y) need not be because of the dependence of y on 8.5

We suppose that 8 is a random variable, with its uncertainty captured by a distribution function F(B) which is assumed to be right continuous and to have strictly positive support on a bounded interval of the real line given by [&8].6 Expected utility is given by:

,j V(U(P, a 4, W(P, I, 0)) dfw. (2.7)

In the sequel, we specify the limits of integration only if they differ from [@,B]. According to our model, the risk associated with variations in 0 is in part like a price risk, since a(p, 0) is a price argument in the indirect utility function, and in part like an income risk operating via the dependence of b(p, .) on 0.’ Since the non-health goods play no further part, we suppress the vector q in the rest of the analysis.

We model a reimbursement insurance scheme as one in which the insured is repaid a fraction c1 of her health expenditures by an insurance company. We rule out a full optimal insurance scheme being implementable by supposing, in line with our discussion above, that whilst the insurance

51n fact the condition for e(p,B,y) to be increasing in 0 is that the demand for health be inelastic, see footnote 12 for further details.

6Hence we are assuming that 8~ CO. ‘The reader who is unfamiliar with the analysis of price risk is referred to Newbery and

Stiglitz (1981).


company may observe 6 it does not know the preferences of the consumer and the health care technology. Hence, it is confronted with observations of 8 and health expenditures and can operate insurance based only on this information. Under the insurance scheme that we envisage, the price vector of health goods faced by an insured individual is (1 -cr)p. We confine consideration to insurance contracts with a rate of insurance between zero and one, which is the same for all goods8 If the insurance premium per unit of insurance is n, then expected utility with insurance is given by:

1 V(( 1 - +(P> 01, WP, 1, ‘3 + b(p, 0) - an) dF(B), (24

where we have made use of the linear homogeneity of a( ., 0) and b( ., 0). The subsequent analysis makes it convenient to introduce the function:

4(& 4 1, n, P) = e(p, 0, ~(4 1-4~~ @, 4, W( 1 - U)P, 1, W), (2.9)

which represents health expenditures allowing for the response in health and health care demand to insurance. The rate of insurance is chosen to maximize expected utility, treating rr as parametric. This yields the first-order condition:

J v,a- l’,+,(b-n)dF=O, (2.10)

omitting the arguments of the functions for the sake of convenience. After using (2.5) we have:

J Vw(( 1-44p> @,I-( 1 - Q(P, 0) - cm - P)(4(4 ~1, I, TC, p) - n) dF = 0. (2.11)

We denote the u which solves (2.11) by c(*. Defining

Rz _ vwww VW ’

(2.12)

we show in the appendix that if R >r,~, then (2.11) has a unique solution with CC*= 1. Hence, the insured chooses complete insurance coverage and the indirect utility function evaluated at the optimal insurance level is independent of 13 and hence may be written as:

‘Justifying restriction to linear insurance contracts raises some interesting issues. Although there are some deductions and limits on conventional insurance arrangements, those which we observe are essentially linear. For an attempt at justifying this, see Holmstrom and Milgrom (1987).

146 T. Besley, Publicly provided disaster insurance

I/(0, Z-n). (2.13)

We are interested in competitive insurance contracts, hence we assume that insurance companies make zero profits in equilibrium. This being so, the insurance premium turns out to be equal to the expected value of health expenditures. Hence,

(2.14)

With IX = LX* = 1, this becomes:’

TX* =I a@, O)y(O, I - TC*) +&I, 0) dF(B). (2.15)

The issue of whether, at equilibrium, an individual would prefer to purchase no insurance to full insurance at the premium n*, is an interesting one. We do not, however, take it further here. We will assume that the equilibrium with c(= 1 is always preferred.

3. Supplementary public insurance

In this section we consider the supplementation of the private insurance scheme described so far, with a limited state insurance scheme, i.e. one which only provides coverage for the worst health states. In doing this we assume that the government can observe the state of health, 8 and the demand for health care. However, as with the insurance company, we assume that it does not know the consumer’s utility function or the health care technology, and. hence cannot implement a fully optimal insurance schedule.’

Whilst the state provides compulsory coverage of very bad outcomes, we do not permit it to have a comparative advantage vis-A-vis the private sector in preventing increased expenditures on health care in response to the incentives which it creates. Hence, the scheme that we have in mind is somewhat analogous to the Medicare system in the United States prior to the introduction of Diagnosis Related Groups.” The state, in effect, contracts out its business to private health care providers. In reality, direct state provision of the health care might allow further controls on the use of health care. We deliberately exclude such a possibility, since we wish to show that state provided insurance may play an important role even if the state

9By this we mean an insurance contract resulting in a stable marginal utility of income and through perfect monitoring of agents’ behavior, prevents adverse incentives resulting in inefficiencies.

“These make insurance payments to hospitals a function of a patient’s medical condition [see Pauly (1986) for a discussion of how they work in practice].


has no direct control on the allocation of health resources.” Hence, if the government sets complete insurance coverage, health expenditures are assumed to be at the same level as they would have been were private insurers to set complete coverage for the same illness.

The analysis that we undertake requires the assumption that

This says that allowing for the fact that a higher 0 will alter the demand for health, total health expenditures are an increasing function of 6. While a plausible requirement at an intuitive level, it may not hold if the demand for health is price elastic in some ranges. I2 Provided, however, that Assumption

4 holds, we can be sure that taking any health state, say 0, we will have higher health expenditures for all tI>e and the converse will be true for O<e. Finally, we also assume:

(3.2)

i.e. that a rise in the premium reduces average health care demand, allowing for a change in a, the amount of insurance coverage chosen. This is a stability condition for the equilibrium in the insurance market.

Suppose now that the government introduces compulsory complete insurance for all illnesses above some level S, financed using a lump-sum tax. The tax required to break even is:

i.e. it is equal to expected expenditures on health care for health states above S. The market for private insurance will be affected by such an arrangement,

“It is a matter of some debate just whether and how state-controlled systems of health care provision are able to improve upon third party insurance schemes. For an interesting discussion of some of the differences between the U.K. and U.S. systems, the reader is referred to Aaron and Schwartz (1984).

“The reader will lind it easy to check that

where B=ar/W’ and E=(+/Ja)/(r/a) holding utility fixed. Hence, provided that the marginal propensity to consume health care is less than one, the demand for health being inelastic is a sufficient condition for this assumption to hold.


since individuals no longer need insurance coverage for illnesses worse than S.r3 Hence, the insurance premium required for private insurance to break even becomes:

7~ =; 6(& &,I, n, P, B) We), (3.4) e

which is the expected value of health expenditures for health states below S. Expected utility is now given by:

~V((l-a)o(p,B),I-(l-a)b(p,~)--an-8)dT(B)

+ V(O,Z -cm-B)[l-F(S)]. (3.5)

This divides into utilities for health states below S and for those above S. The optimal choice of c1 is characterized by:

where we have omitted the arguments of the functions for convenience. We denote the value of CI which satisfies this equation by a**. The following lemma is proved in the appendix and will be useful in the sequel:

Lemma 1. Let a** satisfy (3.6), then

(i) LX** < 1 and

(ii) lims,;a**-+cc*= 1.

The lemma says that after the introduction of public insurance, agents purchase less private insurance coverage and that in the limit as S tends to 0, the rate of private insurance coverage tends to one. We now exhibit conditions under which the introduction of public insurance, as envisaged here, raises welfare.

Proposition. Suppose that

(i) f(8) >O and

13We wish to exclude the possibility that the insured purchases double insurance coverage and hence makes a ‘profit’ for health states above S. Whilst such insurance might be desirable for the insured. it contravenes the notion of a reimbursement contract.

7: Be&y, Publicly provided disaster insurance 149

then there exists a government insurance scheme with S<g which is welfare improving.

Proof. The strategy of proof is to show that expected utility is decreasing in S when S=B. We will show that if (i) and (ii) are satisfied, then a marginal reduction in S below 0 raises expected utility. Differentiating (3.5) with respect to S yields:

dV={V((l-a**)a(p,8),Z-(l-cc**)b(p,8)-a**n**-/I)

-V(O, W-cr**n**-/?)}f(@dS

- ~V,,4(l-c?**)a(~,~),Z--(l-a**)~(p,~)-a**rc**-~)dF(tI) i

+V(O,I-a**n**-j?)[l-F(S)] }{a**$+g}dS. (3.7)

Note that we have used the first-order condition for the choice of a** in deriving (3.7). We need to show that dV >O when dS <O at S= 8. Consider- ing (3.7) as S-+8 yields:

limdV= -jV~((l-a*)a(p,8),1-(l-a*)b(p,61)-a*n*-B)dFo s-8 e

after making use of Lemma 1. The appendix proves:

Lemma 2. Suppose that conditions (i) and (ii) then

of the proposition are satisfied,

(3.8)

This lemma can be used to complete the proof of the proposition since from (3.8) it should be clear that choosing dS <O implies that dV>O at S= 8, after using the fact that the marginal utility of income is positive. Q.E.D.


There is a clear intuitive rationale behind this proposition. The introduction of compulsory complete coverage for health states above some S implies that consumers purchase less private insurance. The gain from the elimina- tion of some measure of moral hazard for states worse than S is outweighed by the costs of its ‘institutionalization’ above this level. Conditions (i) and (ii) guarantee that within a neighborhood of S= 0, the effect upon the insurance premium of a small change in S is defined and positive so that there is a gain (operating via the change in the insurance premium) to the consumer from the introduction of the government insurance scheme.

More specifically, Lemma 2 says that the marginal reduction in the insurance premium outweighs the increase in the lump-sum tax required to finance the government insurance. It is the key to the intuition. Since the consumer is optimizing in the choice of a, the envelope theorem ensures that a small reduction in c1 does not affect expected utility directly. There are, however, two effects. Health states in the interval [Se] are now covered by state insurance. The cost of doing so, which is levied in taxes, is the expected value of health expenditures in this interval (when CI= 1). The private insurance premium is, however, reduced by precisely this amount, so that the tax used to finance state insurance is offset by a reduction in the private insurance premium. There is an additional effect arising from the fact that if private insurance rates are reduced, then so is moral hazard. Hence, the insurance premium in the private sector is reduced still further, and the

insured is better off. The effect of introducing government insurance is to create a non-linearity

in the insurance schedule. The schedule after government intervention is as in fig. 1. From the literature on non-linear pricing it is known that non-linear schemes, where implementable, are, generally speaking, more efficient than

linear ones.14 In practice, the areas in which non-linear pricing is feasible are restricted by the possibility that individuals facing different prices are apt to re-trade the commodity in question. Health insurance and health care are areas in which the scope for non-linear pricing seems more reasonable, however, since such re-trading is not likely to be feasible. Nevertheless, as Pauly (1974) has pointed out, competitive insurance markets are not able to sustain such non-linear pricing schedules since individuals are apt to buy coverage from more than one source.

5. Conclusions

The analysis of this paper is rather piecemeal in the sense that we have explored a particular form of government intervention, rather than seeking to examine the properties of optimal insurance schedules. Neverthless, it is still of interest for a number of reasons. First, the character of the reform that we

14The interested reader is referred to Roberts (1979) for a general discussion.


v w q(s)

Fig. 1

have considered is of the kind which is currently on the policy agenda in the United States and bears upon the types of issues currently being debated in the United Kingdom. Secondly, it highlights the fact that supplementing private insurance coverage by compulsory disaster coverage may have a rationale even if the incentives to consume excessive amounts of health care remain. While direct state control of medical expenses may help to further reduce the demand for medical care, it should be apparent that gains which are possible in this way can be kept distinct from those available through offering insurance against severe outcomes.

A premise of the argument presented here is that private insurers will not offer the kind of insurance scheme which we have shown to be superior to simple linear reimbursement. In justifying this, we appealed to arguments put in Pauly (1974). To the extent that insurance markets diverge from the competitive ideal, there might be less of an argument for the kind of public action considered in this paper since, although separate measures for dealing with the consequences of private monopoly power might have to be

considered. Nevertheless it seems implausible to claim that the schedules which we observe in practice approximate the optimal non-linear schedules analyzed by economic theorists. In view of this, the main idea of this paper - that piecemeal intervention might play an important role - remains a viable one.

Other simpler interventions than that focused upon here might also have a welfare-enhancing role in markets with moral hazard. For example, it can be shown that a tax upon health insurance raises welfare in the present model. The tax treatment of health insurance raises interesting issues.i5 Even with a tax on health insurance in force, the arguments of this paper are not

ISFor a comprehensive survey, see Pauly (1986)


redundant and such a measure might be regarded as complementary with public insurance against very severe outcomes.

In this paper we have provided a rationale for the often made suggestion that state insurance may play a role in combating moral hazard in health insurance. In a model of reimbursement health insurance, we saw that compulsory complete coverage for very bad outcomes may reduce the demand for private insurance to cover other health states and thereby the moral hazard associated with insurance. Such a piecemeal intervention may be of practical relevance and bears upon the current debates about the role of government in the provision of coverage for ill-health. Hence, what we have found has an importance well beyond its theoretical interest.

Appendix

We begin by showing that CY* = 1 is a solution to (2.11). Since rc is equal to expected health expenditures, it is sufficient for (2.11) to be satisfied that V, be independent of 0 at a = 1. Differentiating V,( .) with respect to 0 yields:

This is equal to zero at a= 1 as required. This solution will be the unique solution if the term in parentheses in (A.l) is everywhere positive. The second term is always positive. The sign of the tirst term can be ascertained by differentiating Roy’s Identity, to obtain:

vrV,=P(R-?)Y (A.2)

where R and r~ are defined in the text and /3=ay/ W. Evidently (A.2) is positive, as is the term in parentheses in (A.l), if R>r].16 Hence the claim is proved. 0

Proof of Lemma 1. First we show that a**-+1 as S-+8. Eq. (3.6) can be rewritten as:

(A.31

Since F(S)-+1 as S-+8, the right-hand side of (A.3) tends to zero and we are back to the case described by (2.13). The argument used above to establish that c1= 1 in this case can then be applied. The right-hand side of (A.3) is

16Besley (1989) offers an interpretation of this condition, which arises in a number of contexts, in terms of health being a necessity.

T Besley, Publicly provided disaster insurance 153

positive and under the second-order condition for the choice of a, the left-hand side must be decreasing in c(. Hence, LX** c 1 as claimed. 0

Proof of Lemma 2. In writing down health expenditures we wish to make a distinction between the rate of insurance which is effective at the margin and the rate of private insurance coverage. In states where only private insurance is used, these two will be the same. However, for illnesses above S, the effective rate of insurance will be unity whatever the rate of private insurance. We cope with this by allowing C#J( .) to depend upon both rates of insurance explicitly and denoting this new function by $( *). The effective rate of insurance will be denoted by z whilst we continue to denote the private rate by CC For health states 8<S, we have u=r. In the light of this we define:

and

Note that /?=<(S, 1, ~1, I,rc,/?,p) and rr=x(S,cr,a,Z,rr,/I,p). Differentiating (A.4) with respect to S at z= 1, we have:

(‘4.6)

Note, furthermore, that

lim!!!=!Z s+edS as S-8

= -6(&l, 1,1,~*,0,p)f(Q= -+(Q, l,~,x,p)f(@ (A.7)

after using Lemma 1. Differentiating (A.5) with respect to S with z=o! yields:

drc ax ax dcr ax d/I ax drc dS=$S+dcc.dS+p’dS+j&‘ds.

and, furthermore,

dcr aa aa djI ax dn --_-+-.-+-.- dS as ap dS an dS

(‘4.8)

(A.9)

Substituting (A.9) into (A.8) yields:

154 7: Be&y, Publicly provided disaster insurance

(A.lO)

Define

(A.ll)

Note that lim s_(T Q ~0, after using (3.2). Using (A.1 l), in conjunction with (A.7) yields:

Note for future reference that

‘. (A.12)

(A.13)

Turning now to the expression whose limit we wish to sign, we obtain, by using (A.12) that

+{(l-W’ XM(;$+$.$)+l}.dJ

Using (A.13):

(1 -n)-’ z($+!$)+l+Q)-l.

Hence,

(A.14)

(A.15)

(A.16)

Consider

7: Be&y, Publicly provided disaster insurance 155

= 0, using the fact that lim a** = 1. (A.17) S-R

Hence,

(A.18)

We show finally that under assumptions (i) and (ii) of the lemma, this expression is well defined and positive. Differentiating (3.6) partially with respect to S yields:

d={Vw((l-cc)a(p,S), I-(l-cr)b(p,S)

+ v,(O,I-crn-~)7c}f(S). (A.19)

Provided that the second-order condition is fulfilled, we know that (A.19) will be proportional to &/as. Note that

~~d=T/,(o,W-atn)~(~,l,l,I,rr,O,p)f(iT) (A.20)

after using Lemma 1 again. Hence,

(A.21)

provided that condition (i) of the proposition is fulfilled. Returning to (A.lS), we see that, provided that (1 -Q) > 0, then

lim S-B i

(1 -Q)-l~.z.$ >O, I

(A.22)

since


(A.23)

by condition (ii) of the proposition. 0

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Publicly provided disaster insurance for health and the control of moral hazard